Page 193 - Calculus Workbook For Dummies

Chapter 10

Integration: Reverse Differentiation

In This Chapter
Analyzing the area function
Getting off your fundament (butt) to study the Fundamental Theorem
Guessing and checking
Pulling the switcheroo

n this chapter, you really get into integration in full swing. First you look at the annoying
Iarea function, then the Fundamental Theorem of Calculus, and then two beginner integra-
tion methods.

The Absolutely Atrocious and

Annoying Area Function

The area function is both more difficult and less useful than the material that follows it. With
any luck, your calc teacher will skip it or just give you a cursory introduction to it. Once
you get to the following section on the Fundamental Theorem of Calculus, you’ll have no
more use for the area function. It’s taught because it’s the foundation for the all-important
Fundamental Theorem.

The area function is an odd duck and doesn’t look like any function you’ve ever seen before.
x
h
A x = # ^h f t dt
f ^
s
The input of the function (its argument) is the x on top of the integral symbol. Note that f t ^ h
is not the argument. The output, A xh, tells you how much area has been swept out under
f ^
the curve, f t ^ h, between some starting point, x = s, and the input value. For example, con-
x
sider the simple horizontal line g t = 10 and the area function based on it, A x = # 10 dt.
^ h
g^ h
3
This area function tells you how much area is under the horizontal line between 3 and the
input value. When x = 4, the area is 10 because you’ve got a rectangle with a base of one —
from 3 to 4 — and a height of 10. When x = 5, the output of the function is 20; when x = 6, the
output is 30, and so on. (For an excellent and thorough explanation of the area function and
how it relates to the Fundamental Theorem, check out Calculus For Dummies.) The best way
to get a handle on this weird function is to see it in action, so here goes.
Don’t forget that when using an area function (or a definite integral — stay tuned), area
below the x-axis counts as negative area.